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Journal articles on the topic 'Carathéodory metric'

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Published: 6 September 2023

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1

Fornæss, John Erik, and Lina Lee. "Kobayashi, Carathéodory and Sibony metric." Complex Variables and Elliptic Equations 54, no. 3-4 (March 2009): 293–301. http://dx.doi.org/10.1080/17476930902760450.

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2

Abate, Marco, and Jean-Pierre Vigué. "Isometries for the Carathéodory metric." Proceedings of the American Mathematical Society 136, no. 11 (May 20, 2008): 3905–9. http://dx.doi.org/10.1090/s0002-9939-08-09391-x.

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3

Ge, Zhong. "Collapsing Riemannian Metrics to Carnot-Caratheodory Metrics and Laplacians to Sub-Laplacians." Canadian Journal of Mathematics 45, no. 3 (June 1, 1993): 537–53. http://dx.doi.org/10.4153/cjm-1993-028-6.

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AbstractWe study the asymptotic behavior of the Laplacian on functions when the underlying Riemannian metric is collapsed to a Carnot-Carathéodory metric. We obtain a uniform short time asymptotics for the trace of the heat kernel in the case when the limit Carnot-Carathéodory metric is almost Heisenberg, the limit of which is the result of Beal-Greiner-Stanton, and Stanton-Tartakoff.

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4

CONNELL, CHRIS, THANG NGUYEN, and RALF SPATZIER. "Carnot metrics, dynamics and local rigidity." Ergodic Theory and Dynamical Systems 42, no. 2 (December 9, 2021): 614–64. http://dx.doi.org/10.1017/etds.2021.116.

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AbstractThis paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

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5

Fu, Siqi. "Asymptotic Expansions of Invariant Metrics of Strictly Pseudoconvex Domains." Canadian Mathematical Bulletin 38, no. 2 (June 1, 1995): 196–206. http://dx.doi.org/10.4153/cmb-1995-028-9.

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AbstractIn this paper we obtain the asymptotic expansions of the Carathéodory and Kobayashi metrics of strictly pseudoconvex domains with C∞ smooth boundaries in ℂn. The main result of this paper can be stated as following:Main Theorem. Let Ω be a strictly pseudoconvex domain with C∞ smooth boundary. Let FΩ(z,X) be either the Carathéodory or the Kobayashi metric of Ω. Let δ(z) be the signed distance from z to ∂Ω with δ(z) < 0 for z ∊ Ω and δ(z) ≥ 0 for z ∉ Ω. Then there exist a neighborhood U of ∂Ω, a constant C > 0, and a continuous function C(z,X):(U ∩ Ω) × ℂn -> ℝ such that and|C(z,X)| ≤ C|X| for z ∊ U ∩ Ω and X ∊ ℂn

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6

Krushkal, Samuel. "On the Carathéodory metric of universal Teichmüller space." Ukrainian Mathematical Bulletin 19, no. 1 (January 28, 2022): 75–87. http://dx.doi.org/10.37069/1810-3200-2029-19-1-5.

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In contrast to finite dimensional Teichmuller spaces, all non-expanding invariant metrics on the universal Teichmuller space coincide. This important fact found various applications. We give its new, simplified proof based on some deep features of the Grunsky operator, which intrinsically relate to the universal Teichmuller space. This approach also yields a quantitative answer to Ahlfors' question.

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7

Krushkal, Samuel L. "On the Carathéodory metric of universal Teichmüller space." Journal of Mathematical Sciences 262, no. 2 (April 2022): 184–93. http://dx.doi.org/10.1007/s10958-022-05809-9.

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8

Selivanova, Svetlana. "Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces." Journal of Dynamical and Control Systems 20, no. 1 (December 17, 2013): 123–48. http://dx.doi.org/10.1007/s10883-013-9206-3.

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9

Nikolov, N. "Continuity and boundary behavior of the Carathéodory metric." Mathematical Notes 67, no. 2 (February 2000): 183–91. http://dx.doi.org/10.1007/bf02686245.

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10

Świątkowski, Jacek. "Compact 3-manifolds with a flat Carnot-Carathéodory metric." Colloquium Mathematicum 63, no. 1 (1992): 89–105. http://dx.doi.org/10.4064/cm-63-1-89-105.

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11

Krushkal, Samuel L. "The Grunsky function and Carathéodory metric of Teichmüller spaces." Complex Variables and Elliptic Equations 61, no. 6 (January 11, 2016): 803–16. http://dx.doi.org/10.1080/17476933.2015.1131682.

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12

Martinetti, Pierre. "Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry." Communications in Mathematical Physics 265, no. 3 (April 22, 2006): 585–616. http://dx.doi.org/10.1007/s00220-006-0001-9.

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13

Nikolov, Nikolai, Peter Pflug, Pascal J. Thomas, and Włodzimierz Zwonek. "Estimates of the Carathéodory metric on the symmetrized polydisc." Journal of Mathematical Analysis and Applications 341, no. 1 (May 2008): 140–48. http://dx.doi.org/10.1016/j.jmaa.2007.09.072.

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14

Arstu and Swadesh Kumar Sahoo. "Carathéodory Density of the Hurwitz Metric on Plane Domains." Bulletin of the Malaysian Mathematical Sciences Society 43, no. 6 (April 29, 2020): 4457–67. http://dx.doi.org/10.1007/s40840-020-00937-4.

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15

Tan, Kang-Hai, and Xiao-Ping Yang. "Characterisation of the sub-Riemannian isometry groups of the H-type groups." Bulletin of the Australian Mathematical Society 70, no. 1 (August 2004): 87–100. http://dx.doi.org/10.1017/s000497270003584x.

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For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesics. We use properties of sub-Riemannian geodesics in G to characterise the isometry group ISO(G) with respect to the Carnot-Carathéodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.

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16

Karmanova, M. B. "Local Metric Properties of Level Surfaces on Carnot–Carathéodory Spaces." Doklady Mathematics 99, no. 1 (January 2019): 75–78. http://dx.doi.org/10.1134/s1064562419010241.

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17

Wang, Jianfei. "Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball." Abstract and Applied Analysis 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/647972.

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LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

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18

Skrzypczak, Leszek. "Besov Spaces and Hausdorff Dimension For Some Carnot-Carathéodory Metric Spaces." Canadian Journal of Mathematics 54, no. 6 (December 1, 2002): 1280–304. http://dx.doi.org/10.4153/cjm-2002-049-x.

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AbstractWe regard a system of left invariant vector fields satisfying the Hörmander condition and the related Carnot-Carathéodory metric on a unimodular Lie group G. We define Besov spaces corresponding to the sub-Laplacian both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

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19

Le Donne, Enrico. "A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries." Analysis and Geometry in Metric Spaces 5, no. 1 (January 4, 2018): 116–37. http://dx.doi.org/10.1515/agms-2017-0007.

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Abstract Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.

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20

Abate, Marco, and Roberto Tauraso. "The Lindelöf principle and angular derivatives in convex domains of finite type." Journal of the Australian Mathematical Society 73, no. 2 (October 2002): 221–50. http://dx.doi.org/10.1017/s1446788700008818.

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AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

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21

Klein, Tom, and Andrew Nicas. "The Horofunction boundary of the Heisenberg Group: The Carnot-Carathéodory metric." Conformal Geometry and Dynamics of the American Mathematical Society 14, no. 15 (November 17, 2010): 269. http://dx.doi.org/10.1090/s1088-4173-2010-00217-1.

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22

BALOGH, ZOLTÁN M., RETO BERGER, ROBERTO MONTI, and JEREMY T. TYSON. "Exceptional sets for self-similar fractals in Carnot groups." Mathematical Proceedings of the Cambridge Philosophical Society 149, no. 1 (March 24, 2010): 147–72. http://dx.doi.org/10.1017/s0305004110000083.

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AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

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23

Karmanova, M. B. "Metric Properties of Graphs on Carnot–Carathéodory Spaces with Sub-Lorentzian Structure." Doklady Mathematics 101, no. 1 (January 2020): 36–39. http://dx.doi.org/10.1134/s1064562420010159.

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24

Gekhtman, Dmitri, and Vladimir Markovic. "Classifying complex geodesics for the Carathéodory metric on low-dimensional Teichmüller spaces." Journal d'Analyse Mathématique 140, no. 2 (March 2020): 669–94. http://dx.doi.org/10.1007/s11854-020-0102-y.

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25

Ruszkowski, Bartosch. "Hardy inequalities for the Heisenberg Laplacian on convex bounded polytopes." MATHEMATICA SCANDINAVICA 123, no. 1 (August 1, 2018): 101–20. http://dx.doi.org/10.7146/math.scand.a-105218.

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We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carathéodory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.

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26

Magnani, Valentino. "On a measure-theoretic area formula." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 4 (July 20, 2015): 885–91. http://dx.doi.org/10.1017/s030821051500013x.

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We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory measure. We focus our attention on the case when this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our aim is to use certain covering derivatives as ‘generalized densities’. Some consequences for the sub-Riemannian Heisenberg group are also pointed out.

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27

Wang, Huiju, and Pengcheng Niu. "Local boundedness for minimizers of convex integral functionals in metric measure spaces." MATHEMATICA SCANDINAVICA 126, no. 2 (May 6, 2020): 259–75. http://dx.doi.org/10.7146/math.scand.a-116244.

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In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

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28

KLOEDEN, PETER E., and VICTOR S. KOZYAKIN. "ASYMPTOTIC BEHAVIOUR OF RANDOM MARKOV CHAINS WITH TRIDIAGONAL GENERATORS." Bulletin of the Australian Mathematical Society 87, no. 1 (March 30, 2012): 27–36. http://dx.doi.org/10.1017/s0004972712000160.

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AbstractContinuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carathéodory random differential equations and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself with respect to the Hilbert projective metric. It does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transition probabilities, in which case the attractor is a periodic path.

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29

Han, Yongsheng, Detlef Müller, and Dachun Yang. "A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces." Abstract and Applied Analysis 2008 (2008): 1–250. http://dx.doi.org/10.1155/2008/893409.

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We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces whenp≥1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains inℂN. Our theory applies in a wide range of settings.

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30

Dlugie, Ethan, and Aaron Peterson. "On uniform large-scale volume growth for the Carnot–Carathéodory metric on unbounded model hypersurfaces in ℂ2." Involve, a Journal of Mathematics 11, no. 1 (January 1, 2018): 103–18. http://dx.doi.org/10.2140/involve.2018.11.103.

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31

Ikonen, Toni. "Quasiconformal Jordan Domains." Analysis and Geometry in Metric Spaces 9, no. 1 (January 1, 2021): 167–85. http://dx.doi.org/10.1515/agms-2020-0127.

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Abstract We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY ). We say that a metric space (Y, dY ) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY ) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.

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32

Flynn, Joshua, Nguyen Lam, and Guozhen Lu. "Sharp Hardy Identities and Inequalities on Carnot Groups." Advanced Nonlinear Studies 21, no. 2 (March 12, 2021): 281–302. http://dx.doi.org/10.1515/ans-2021-2123.

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Abstract In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

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33

Mitchell, John. "On Carnot-Carathéodory metrics." Journal of Differential Geometry 21, no. 1 (1985): 35–45. http://dx.doi.org/10.4310/jdg/1214439462.

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34

Agler, Jim, Zinaida Lykova, and N. J. Young. "Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc." Journal of Geometric Analysis 31, no. 8 (January 19, 2021): 8202–37. http://dx.doi.org/10.1007/s12220-020-00582-0.

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AbstractThe symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

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35

Ge, Zhong. "Horizontal path spaces and Carnot-Carathéodory metrics." Pacific Journal of Mathematics 161, no. 2 (December 1, 1993): 255–86. http://dx.doi.org/10.2140/pjm.1993.161.255.

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36

Hamenstädt, Ursula. "Some regularity theorems for Carnot-Carathéodory metrics." Journal of Differential Geometry 32, no. 3 (1990): 819–50. http://dx.doi.org/10.4310/jdg/1214445536.

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37

Mahajan, Prachi. "On isometries of the Kobayashi and Carathéodory metrics." Annales Polonici Mathematici 104, no. 2 (2012): 121–51. http://dx.doi.org/10.4064/ap104-2-2.

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38

Dontsov, V. V. "Systoles on Heisenberg groups with Carnot-Carathéodory metrics." Sbornik: Mathematics 192, no. 3 (April 30, 2001): 347–74. http://dx.doi.org/10.1070/sm2001v192n03abeh000549.

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39

Krantz, Steven G. "The Carathéodory and Kobayashi Metrics and Applications in Complex Analysis." American Mathematical Monthly 115, no. 4 (April 2008): 304–29. http://dx.doi.org/10.1080/00029890.2008.11920531.

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40

Greshnov, A. V. "Metrics and tangent cones of uniformly regular Carnot—Carathéodory spaces." Siberian Mathematical Journal 47, no. 2 (March 2006): 209–38. http://dx.doi.org/10.1007/s11202-006-0036-3.

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41

Mahajan, Prachi, and Kaushal Verma. "Some Aspects of the Kobayashi and Carathéodory Metrics on Pseudoconvex Domains." Journal of Geometric Analysis 22, no. 2 (December 4, 2010): 491–560. http://dx.doi.org/10.1007/s12220-010-9206-4.

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42

Royden, Halsey, Pit-Mann Wong, and Steven G. Krantz. "The Carathéodory and Kobayashi/Royden metrics by way of dual extremal problems." Complex Variables and Elliptic Equations 58, no. 9 (September 2013): 1283–98. http://dx.doi.org/10.1080/17476933.2012.662226.

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43

Peterson, Aaron. "Carnot–Carathéodory metrics in unbounded subdomains of $${{\mathbb{C}}^2}$$ C 2." Archiv der Mathematik 102, no. 5 (May 2014): 437–47. http://dx.doi.org/10.1007/s00013-014-0646-0.

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44

Markovic, Vladimir. "Carathéodory’s metrics on Teichmüller spaces and $L$ -shaped pillowcases." Duke Mathematical Journal 167, no. 3 (February 2018): 497–535. http://dx.doi.org/10.1215/00127094-2017-0041.

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45

FOURNIER, RICHARD, and STEPHAN RUSCHEWEYH. "A generalization of the Schwarz–Carathéodory reflection principle and spaces of pseudo-metrics." Mathematical Proceedings of the Cambridge Philosophical Society 130, no. 2 (March 2001): 353–64. http://dx.doi.org/10.1017/s0305004100004941.

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46

Bieske, Thomas, and Luca Capogna. "The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics." Transactions of the American Mathematical Society 357, no. 2 (September 23, 2004): 795–823. http://dx.doi.org/10.1090/s0002-9947-04-03601-3.

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47

Capogna, Luca, and Nicola Garofalo. "Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for carnot-carathéodory metrics." Journal of Fourier Analysis and Applications 4, no. 4-5 (July 1998): 403–32. http://dx.doi.org/10.1007/bf02498217.

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48

Wong, Kwok-Kin, and Sai-Kee Yeung. "Quasi-Projective Manifolds Uniformized by Carathéodory Hyperbolic Manifolds and Hyperbolicity of Their Subvarieties." International Mathematics Research Notices, June 27, 2023. http://dx.doi.org/10.1093/imrn/rnad134.

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Abstract Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré-type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.

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49

Sarkar, Amar Deep, and Kaushal Verma. "A submultiplicative property of the Carathéodory metric on planar domains." Proceedings - Mathematical Sciences 130, no. 1 (June 6, 2020). http://dx.doi.org/10.1007/s12044-020-00565-9.

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Hajłasz, Piotr, and Scott Zimmerman. "Geodesics in the Heisenberg Group." Analysis and Geometry in Metric Spaces 3, no. 1 (October 29, 2015). http://dx.doi.org/10.1515/agms-2015-0020.

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Abstract:

Abstract We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.

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